# Integrals over chained functions

SchroedingersLion
Good evening!

Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part.

Suppose we have a concatenated function, say the fermi function ##f(\epsilon)## that goes from R to R. We know that ##\epsilon## is a function of the 3 dimensional wave vector k . If there are applied electromagnetic fields, the wave vector depends on time t', i.e. k(t').

Now suppose we want to calculate an integral $$\int_{-\infty}^t p(t')\frac {df}{dt'} \, dt'$$ with some scalar function ##p(t')##.
Chain rule tells us that ##\frac {df}{dt'} = \frac {df}{d\epsilon} \frac {d\epsilon}{d\mathbf k}\frac {d\mathbf k}{dt'}##, so we get $$\int_{-\infty}^t p(t')\frac {df}{d\epsilon} \frac {d\epsilon}{d\mathbf k}\frac {d\mathbf k}{dt'} \, dt'$$

So, assuming that ##\frac {df}{d\epsilon}## would still depend on ##\epsilon ## and ##\frac {d\epsilon}{d\mathbf k}## would still depend on ##\mathbf k##, am I right in assuming that both of these terms would also be considered as being dependent on t'?
Because ##\epsilon## depends on ##\mathbf k## and the latter on t'. So, under the integral, I would express ##\epsilon## and ##\mathbf k## in terms of t', before I integrate.

Yet, the book keeps acting as if the time dependence only comes from ##p(t)## and ##\frac {d\mathbf k}{dt'}##.

Best,
SL